Suppose there are two overlapping overlapping point sets in \(\mathbb {R}^m\) , the model shape \(Q \Delta \over = \lbrace  {\vec{q}_i}\rbrace  _{i = 1}^{{N_q}}\)  and a data shape \(P \Delta \over = \lbrace  {\vec{p}_i}\rbrace  _{i = 1}^{{N_p}}\)  \(({N_p},{N_q} \in \mathbb {N})\) , where \(\xi \)  represents the overlapping percentage of the data shape, \({P_\xi }\)  denotes the point subset, which is the overlapping part of the data shape to the model shape and \(N_p^{^{\prime }}\)  indicates the number of points in \({P_\xi }\) . According to the idea of the trimmed ICP algorithm [1]}, the optimal scaling transformation can be obtained by minimizing the following objective function:
\(\Psi (\xi ,s,{\bf {R}},\vec{t}) = \frac{{e(\xi ,s,{\bf {R}},\vec{t})}}{{{s^2} \cdot {\xi ^{1 + \lambda }}}},\) 
